Common Core Standards Addressed in this Resource

Transcription

1 Common Core Standards Addressed in this Resource N-CN.4 - Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. Activity pages: 6, 7, 8 F-IF.7 - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Activity pages: 2, 24 F-TF. - Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. Activity pages:, 4 F-TF.2 - Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. Activity pages: 7, 0 F-TF.5 - Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline. Activity pages:, 20, 2, 22 F-TF.7 - Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context. Activity pages: 4, 7, 8 F-TF.8 - Prove the Pythagorean identity sin2(θ) + cos2(θ) = and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. Activity pages: 7, 8 F-TF. - Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Activity pages: 5, 6, 26 G-SRT.6 - Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Activity pages: 5,,, 2 G-SRT.8 - Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Activity pages: 6, 8, G-SRT. - Derive the formula A = /2 ab sin(c) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the side. Activity page: 2 G-SRT. - Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). Activity pages: 0,,, 4

2 Table of Contents Measures of Angles Arc Length Trigonometric Functions of Acute Angles Applying Trigonometry in Right Triangles Redefining the Trigonometric Functions Using One Function Value to Find Others Representing Trigonometric Functions as Line Segments Function Values of Quadrantal Angles Function Values of 0, 45, Function Values of Angles of Any Size Representing One Trigonometric Function in Terms of Another Inverse Trigonometric Notation Solving First-Degree Trigonometric Equations More Equations with First-Degree Trigonometric Functions Solving Second-Degree Trigonometric Equations Graphs of Sine and Cosine Functions Graphs of Other Trigonometric Functions Graphing Systems of Trigonometric Equations Basic Trigonometric Identities Sum and Difference Identities Double-Angle Identities Half-Angle Identities The Ambiguous Case The Law of Sines The Law of Cosines Area of a Triangle Parallelogram of Forces Navigation Problems Literal Triangle Problems Polar Coordinates Polar Graphs Trigonometric Form of Complex Numbers Multiplication and Division in Trigonometric Form Assessment A: Trigonometric Functions Assessment B: Graphs of Trigonometric Functions Assessment C: Trigonometric Equations and Identities Assessment D: Solving Triangles Answer Key Graphs of Compound Trigonometric Functions

3 Name Measures of Angles. An angle is formed by two rays with a common endpoint. The size of an angle is the amount of rotation between its two rays. 2. Counterclockwise rotation is positive. Clockwise rotation is negative.. Units for measuring rotation are revolution, degree, radian, and grad (gradian or gradient). 4. Conversion Formulas: revolution = 60 π radians = grads = 0 Terminal Side θ Initial Side Initial Side θ Terminal Side Example: π radians π Convert 60 to radians. 60 x = radians 80 Play odd measure out. Cross through the measure in each row that is not equivalent to the other three.. revolution 60 π radians 400 grads 2. π radians revolution 66 2 grads π 2 grads radians revolution grads 7π radians 7 revolution 2 4π radians 2 revolutions grads 6.,200 grads,080 revolutions 6 radians 7. 2π revolution radians 540 π grad π 600 Milliken Publishing Company MP50

4 Name The linear measure, s, of an arc of a circle is related to the radian measure of its central angle, θ, and the radius, r. s = θ r θ r s Arc Length Example: Find the length of the arc that subtends an angle of 40 in a circle whose radius is 8 inches. Answer to the nearest tenth of an inch.. Convert the angle measure from degrees to radians. 2. Substitute the radian value for θ and the value of the radius into the arc-length formula. π radians 40 x = 80 s = θr s = 2 π x 8 2π radians. Substitute for π, and evaluate. 24 (. ) s x inches Find the indicated measures. Then use the answer code to complete the seven-word sentence below.. In a circle with radius 2 centimeters, find the length of an arc intercepted by a central angle of A circle has a radius of 6 feet. Find the radian measure of a central angle that intercepts an arc length of 2 feet.. In a circle, a central angle of 0 intercepts an arc of 2.5 inches. Find the length of the radius. 4. The length of a pendulum is 8 inches. Find the distance through which the tip of the pendulum travels when the pendulum turns through an arc of 2.5 radians. 5. The diameter of a wheel is 48 inches. Find the number of degrees through which a point on the circumference turns when the wheel moves a distance of 2 feet theta 57. pi 44. in. could 45 in. calculate 2π would 2 wish 0.5 recalculate π cm how cm think I I. Count the letters in each of the seven words. Use the count to fill in the spaces below, revealing the values of the first seven digits of π. Milliken Publishing Company 4 MP50

5 Name Trigonometric Functions of Acute Angles. There are 6 trigonometric functions: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot) 2. Cofunctions: sine and cosine, tangent and cotangent, secant and cosecant. Reciprocal Functions: (sine, cosecant), (cosine, secant), (tangent, cotangent) 4. In a right triangle, the is always the right angle. In right triangle ABC, legs a and b are named with respect to acute angle A. sin A = a, csc A = c c a A cos A = b, sec A = c c b b c tan A = a, cot A = b b a C a B Across. sin A = B 5. relationship between the 2 acute angles of a right triangle 4 6. sin A = A 5. A = cot B Down A = 2 4. cos A side right angle 7. product of two reciprocals. A = tan A 8. Milliken Publishing Company 5 MP50